A noncommutative probability theory

Abstract

A noncommutative probability theory is developed in which no boundedness, finiteness, or “tracial” conditions are imposed. The underlying structure of the theory is a “probability algebra” ( a , ω ) (\mathcal {a},\omega ) where a \mathcal {a} is a *-algebra and ω \omega is a faithful state on a \mathcal {a} . Conditional expectations and coarse-graining are discussed. The bounded and unbounded commutants are considered and commutation theorems are proved. Two classes of probability algebras, which we call closable and symmetric probability algebras are shown to have important regularity properties. The canonical algebra of quantum mechanics is considered in some detail and a strong commutation theorem is proven for this case. Moreover, in this case, isotropic normal states, KMS states, and stable states are defined and characterized.